partially ordered group
3) for all .
If, in a group , one can find a set with the properties 1)–3), then can be made into a -group by setting if and only if . It is correct to identify the order of a -group with its positive cone. One often writes a -group with positive cone as .
A mapping from a -group into a -group is an order homomorphism if is a homomorphism of the group and for all ,
A homomorphism from a -group into a -group is an order homomorphism if and only if .
A subgroup of a -group is called convex (cf. Convex subgroup) if for all with ,
If is a convex subgroup of a -group , then the set of right cosets of by is a partially ordered set with the induced order if there exists an such that . The quotient group of a -group by a convex normal subgroup is a -group respect with the induced partial order, and the natural homomorphism is an order homomorphism. The homomorphism theorem holds for -groups: if is an order homomorphism from a -group into a -group , then the kernel of is a convex normal subgroup of and there exists an order isomorphism from the -group into such that .
This article extends and updates the article Partially ordered group (Volume 7).
|[a1]||L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)|
Po-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Po-group&oldid=11826